Page 56 - Applied statistics and probability for engineers

34 Chapter 2/Probability

measured to monitor the reaction. Suppose that the equiva- 2-70. Disks of polycarbonate plastic from a supplier are
lence point is reached after approximately 100 mL of an NaOH analyzed for scratch and shock resistance. The results from 100
solution has been added (enough to react with all the acetic disks are summarized as follows:
acid present) but that replicates are equally likely to indicate
from 95 to 104 mL to the nearest mL. Assume that volumes Shock Resistance
are measured to the nearest mL and describe the sample space. High Low
(a) What is the probability that equivalence is indicated Scratch High 70 9
at 100 mL? Resistance Low 16 5
(b) What is the probability that equivalence is indicated at less
than 100 mL? Let A denote the event that a disk has high shock resistance,
(c) What is the probability that equivalence is indicated and let B denote the event that a disk has high scratch resist-
between 98 and 102 mL (inclusive)? ance. If a disk is selected at random, determine the following
probabilities:
2-65. In a NiCd battery, a fully charged cell is composed of (a) P A ( ) (b) P B ( ) (c) P A′ ( )
nickelic hydroxide. Nickel is an element that has multiple oxida- ( B) ( B) (f) P A′ ( B)
tion states and that is usually found in the following states: (d) P A∩ (e) P A∪
2-71. Samples of emissions from three suppliers are classiied
Nickel Charge Proportions Found
for conformance to air-quality speciications. The results from
0 0.17
100 samples are summarized as follows:
+2 0.35
+3 0.33 Conforms
+4 0.15 Yes No
1 22 8
(a) What is the probability that a cell has at least one of the
positive nickel-charged options? Supplier 2 25 5
(b) What is the probability that a cell is not composed of a 3 30 10
positive nickel charge greater than +3? Let A denote the event that a sample is from supplier 1, and let B
2-66. A credit card contains 16 digits between 0 and 9. How- denote the event that a sample conforms to speciications. If a sam-
ever, only 100 million numbers are valid. If a number is entered ple is selected at random, determine the following probabilities:
randomly, what is the probability that it is a valid number? (a) P A( ) (b) P B( ) (c) P A( )′
2-67. Suppose your vehicle is licensed in a state that (d) P A( ∩ B) (e) P A( ∪ B) (f) P A( ′ ′ B)
issues license plates that consist of three digits (between 0 and
9) followed by three letters (between A and Z). If a license 2-72. An article in the Journal of Database Management
[“Experimental Study of a Self-Tuning Algorithm for DBMS
number is selected randomly, what is the probability that yours
Buffer Pools” (2005, Vol. 16, pp. 1–20)] provided the workload
is the one selected?
used in the TPC-C OLTP (Transaction Processing Performance
2-68. A message can follow different paths through servers on
Council’s Version C On-Line Transaction Processing) bench-
a network. The sender’s message can go to one of ive servers
mark, which simulates a typical order entry application. See
for the irst step; each of them can send to ive servers at the
Table 2E-1.
second step; each of those can send to four servers at the third
The frequency of each type of transaction (in the second
step; and then the message goes to the recipient’s server.
column) can be used as the percentage of each type of trans-
(a) How many paths are possible?
action. The average number of selects operations required for
(b) If all paths are equally likely, what is the probability that
each type of transaction is shown. Let A denote the event of
a message passes through the irst of four servers at the
transactions with an average number of selects operations of 12
third step?
or fewer. Let B denote the event of transactions with an aver-
2-69. Magnesium alkyls are used as homogenous catalysts in the
age number of updates operations of 12 or fewer. Calculate the
production of linear low-density polyethylene (LLDPE), which
following probabilities.
requires a iner magnesium powder to sustain a reaction. Redox
reaction experiments using four different amounts of magnesium (a) P A( ) (b) P B( ) (c) P A( ∩ B)
′
powder are performed. Each result may or may not be further (d) P A( ∩ B ) (e) P A( ∪ B)
reduced in a second step using three different magnesium powder 2-73. Use the axioms of probability to show the following:
amounts. Each of these results may or may not be further reduced (a) For any event E,P E ( ) = − (
′
P E).
1
in a third step using three different amounts of magnesium powder.
(b) P ∅ ( ) = 0
(a) How many experiments are possible?
P B).
(b) If all outcomes are equally likely, what is the probability (c) If A is contained in B, then P A ( ) ≤ (
that the best result is obtained from an experiment that uses 2-74. Consider the endothermic reaction’s in Exercise 2-50. Let
all three steps? A denote the event that a reaction's inal temperature is 271 K or
(c) Does the result in part (b) change if ive or six or seven dif- less. Let B denote the event that the heat absorbed is above target.
ferent amounts are used in the irst step? Explain. Determine the following probabilities.